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Ghosttones and LucyTuning

🔗Matt Nathan <mattn@...>

2/19/1997 3:11:42 AM
PAULE wrote:
>
> Matt, I pursued the entire line of discussion with Charles
> Lucy years ago on rec.music.compose. The man is clearly
> interested only in promulgating his opinions -- any factual
> evidence that "supports" them he will gladly jump on, while
> all contradictory evidence or semblance of mathematical reasoning
> he evades through obfuscatory ramblings.

My efforts may be futile. Oh well. If anybody wants
this thread truncated, let me know.

> ...
> Despite all this, LucyTuning is about the best tuning around for the
> existing diatonic, triadic repertoire. Let me try to explain why.
>
> First, it is a meantone tuning. This means that 5-limit just
> intonation is approached without any commas to deal with.

> For
> anyone who denies the importance of this, take a piece of
> diatonic, triadic music and try to render it in just
> intonation. Inevitably, different-sized whole tones, comma
> shifts during sustained notes, out-of-tune chords, and/or
> wandering tonics arise.

I'd disagree here.

Different sized whole tones are not a problem, but rather
a revelation. Their confounding into a single interval
loses a beautiful and natural distinction. The perfect
example is the 8:9:10:12 voicing.

Comma shifts are not a "problem" either, since they are
required by the harmonic context and make perfect sense
as in-tune chord members. Comma shifts are merely voice
leading, in the same manner as one would move by seconds
to voice lead--the only difference is the smaller size of
commas (such as my favorite 81/80) as compared to seconds.
To say that comma shifts are somehow invalid is like saying
that it's invalid to move the B up to a C when resolving
V to I in C. Voice leading is voice leading.

Out-of-tune chords only happen in JI when you don't use
the correct pitches, as when your instrument's pitch set
is finite and you must employ the closest available
substitute. This is not true JI but a form of "surdism"
(a coined modification of a term I read in Partch).

"Wandering tonics" is a misnomer. Wandering tonics fall
into the voice-leading category, as demanded by harmonic
context. A vi minor chord whose third is not the same pitch
as the tonic of the key is not a relative of nor a substitute
for the I chord--it's a different chord with its own
required pitches. (There is also a vi minor chord which is
a relative of the I chord, of course.) The fact that
the third of this chord lies close to the tonic of the key
should not be taken to mean that they are both tonics of
the key, one being "wandered", any more than the leading
tone should be considered a "wandered" tonic, or the major
third of the V of ii (near C# in C) should be considered a
"wandered" tonic.

I think the "problems" some people have in trying to
translate existing "diatonic triadic" music into JI
are related to conceptions about pitch class. I hold
that there are more than 7 pitch classes implied in
much supposedly diatonic music, but that many of the
pitch classes are confounded into single representation
in 12tet and other finite tuning systems. These systems
allow for errors to be written into the music without
being caught because the errors are distributed across some
number of chords in a progression. When such music is
"translated" into JI the errors become apparent and JI
is blamed (like killing the messenger of bad news)
rather than the confounded tuning system which allowed
the composer to write those errors in the first place.

> ...if you take all 5-limit
> just intervals and calculate the meantone tuning that
> minimizes the mean squared error (which is appropriate if you
> think dissonance is a function whose first derivative is zero
> and second derivative is positive at just intervals (I do)),

Will you explain this dissonance function with
figures please? It sounds interesting but I'm
not sure what you mean by derivative.

In JI, I prefer to think of the "complexity" (or something,
rather than "dissonance") of an interval or sonority, which
I measure not by multiplying the members of a ratio or
chord, but by adding them! As an example, 4/1 is about as
"complex" as 3/2 by my rule of thumb, and 8/5 is about as
"complex" as 9/4.

In systems which deviate from just by small amounts,
I'm not really sure how to calculate dissonance,
except that it may have something to do with the
speed of beating (faster beating, more dissonance).

> you get a perfect fifth of
> 696.165 cents. If you use a weighted mean, with weights
> inversely proportional to the limit (following Partch
> who said higher-limit intervals need to be tuned more
> accurately in proportion to their limit), you get a
> perfect fifth of 696.019 cents. The optimal tuning for
> any real musical situation (as long as it is diatonic
> and triadic) will probably fall somewhere between these
> two tunings (a very narrow range!).

That's very interesting mathematically and I will want
to work it through from scratch myself someday
to really "get it". I'm afraid I'd still classify
weighted mean-tone as an error-spreading and pitch-class-
confounding system though. Its main value would be
in the tuning of instruments whose pitch sets are finite
and not manually adjustable during performance, especially
for period music.

> Lucy is thus doing considerably better than 19-equal, and
> about as well as 31-equal. It's no wonder that an eminent
> scholar of days past (John Harrison) advocated the pi-based
> tuning (LucyTuning).

Of course this is not the same as claiming to have found a
better physical model for the mapping of ghosttones as
Charles Lucy is.

Matt Nathan

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